This triangle in the image is isosceles, as indicated by the two equal sides marked with dashes (each 12 cm long). We are tasked with finding the height $b$, and we know one of the angles is $27^\circ$. The height forms a right triangle by splitting the isosceles triangle into two smaller right triangles. The $27^\circ$ angle refers to the angle between the height and the equal side.

We can use trigonometry, specifically the sine function, to find the height $b$. In the right triangle, the sine of $27^\circ$ is the ratio of the opposite side (the height $b$) to the hypotenuse (which is 12 cm).

The equation is:

$\sin(27^\circ) = \frac{b}{12}$

To find $b$, we rearrange the equation:

$b = 12 \times \sin(27^\circ)$

Let me calculate that for you.The height $b$ of the triangle is approximately $5.45$ cm.

Would you like further details or explanations?

Here are some related questions to explore:

1. How can we calculate the base of the triangle using the known angle?
2. What is the total area of the triangle?
3. Can cosine or tangent be used to solve for any other lengths?
4. How does the triangle's symmetry affect the solution?
5. What would change if the angle were increased or decreased?

Tip: In an isosceles triangle, the height always bisects the base, forming two congruent right triangles. This symmetry can simplify many calculations!